第三課時(shí) 最值與范圍

第三課時(shí)最值與范圍題型一不等式法求最值、范圍例1(2024·海南診斷)已知橢圓C：eq\f(x2,a2)＋eq\f(y2,b2)＝1(a>b>0)的左、右焦點(diǎn)分別為F1，F(xiàn)2，P是橢圓C上的一個(gè)動(dòng)點(diǎn)，|PF1|＋|PF2|＝eq\f(\r(6),2)|F1F2|，當(dāng)PF2⊥x軸時(shí)，|PF2|＝eq\f(\r(6),3).(1)求C的方程；(2)設(shè)點(diǎn)P在第一象限，且直線(xiàn)PF1，PF2與橢圓C分別相交于另外兩點(diǎn)A(x1，y2)和B(x2，y2)，求y1－y2的最大值．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________感悟提升利用不等式法求解最值、范圍問(wèn)題的策略(1)利用圓錐曲線(xiàn)的幾何性質(zhì)或幾何量之間的關(guān)系構(gòu)造不等式．(2)利用直線(xiàn)與圓錐曲線(xiàn)的位置關(guān)系、判別式構(gòu)造不等式．(3)利用已知或其他隱含條件中的不等關(guān)系構(gòu)造不等式．(4)常與一元二次不等式、基本不等式相關(guān)．訓(xùn)練1已知F(eq\r(3)，0)是橢圓C：eq\f(x2,a2)＋eq\f(y2,b2)＝1(a>b>0)的一個(gè)焦點(diǎn)，點(diǎn)Meq\b\lc\(\rc\)(\a\vs4\al\co1(\r(3)，\f(1,2)))在橢圓C上．(1)求橢圓C的方程；(2)若直線(xiàn)l與橢圓C相交于A，B兩點(diǎn)，且kOA＋kOB＝－eq\f(1,2)(O為坐標(biāo)原點(diǎn))，求直線(xiàn)l的斜率的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________題型二函數(shù)法求最值、范圍例2(2024·深圳模擬)已知拋物線(xiàn)E：y2＝2px(p>0)的焦點(diǎn)為F，點(diǎn)M(4，m)在拋物線(xiàn)E上，且△OMF的面積為eq\f(1,2)p2(O為坐標(biāo)原點(diǎn))．(1)求拋物線(xiàn)E的方程；(2)過(guò)焦點(diǎn)F的直線(xiàn)l與拋物線(xiàn)E交于A，B兩點(diǎn)，過(guò)A，B分別作垂直于l的直線(xiàn)AC，BD，分別交拋物線(xiàn)于C，D兩點(diǎn)，求|AC|＋|BD|的最小值．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________感悟提升利用函數(shù)法求解最值、范圍問(wèn)題的策略(1)引入單變量，將所求解問(wèn)題用含該變量的代數(shù)式表示，構(gòu)造函數(shù)；(2)引入雙變量，利用題設(shè)或其他條件建立兩個(gè)變量之間的關(guān)系，通過(guò)消元的方法轉(zhuǎn)化為單變量問(wèn)題，構(gòu)造函數(shù)；(3)注意挖掘變量所滿(mǎn)足的條件，從而確定變量范圍；(4)用函數(shù)的方法，如函數(shù)的單調(diào)性、導(dǎo)數(shù)等分析求解最值或范圍．訓(xùn)練2(2023·全國(guó)甲卷)已知直線(xiàn)x－2y＋1＝0與拋物線(xiàn)C：y2＝2px(p>0)交于A，B兩點(diǎn)，|AB|＝4eq\r(15).(1)求p；(2)設(shè)F為C的焦點(diǎn)，M，N為C上兩點(diǎn)，且eq\o(FM,\s\up6(→))·eq\o(FN,\s\up6(→))＝0，求△MFN面積的最小值．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________